related topics {math, number, function}

In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C.

(Recall that a category C is preadditive if all its hom-sets are Abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of Abelian groups. Recall also that a biproduct in a preadditive category is both a finite product and a finite coproduct.)

Warning: The term "additive category" is sometimes applied to any preadditive category, but Wikipedia does not follow this older practice.

## Contents

### Definition

A category C is additive if

• it has a zero object
• every hom-set Hom(A,B) has an addition, endowing it with the structure of an abelian group, and such that composition of morphisms is bilinear
• all finite biproducts exist.

Note that a category is called preadditive if just the second holds, whereas it is called semiadditive if both the first and the third hold.

Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.

### Examples

The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given point-wise, and biproducts are given by direct sums.

More generally, every module category over a ring R is additive, and so in particular, the category of vector spaces over a field K is additive.